# Jacobian-aligned infill for black-box neural tuning

- ID: 80
- Canonical URL: https://synthcore.org/idea/80/jacobian-aligned-infill-for-black-box-neural-tuning
- API JSON: https://synthcore.org/api/idea/80.json
- API Markdown: https://synthcore.org/api/idea/80.md
- Verification status: beats_baseline
- Source: [arXiv:2608.22862](https://arxiv.org/abs/2608.22862)
- Category: optimization
- Solves: sample-efficiency, speedup, accuracy
- ML areas: optimizer, fine-tuning, quantization, scheduler
- Math tags: optimization, linear-algebra, numerical-analysis
- Ratings: usefulness 6/10; difficulty 5/10; novelty 7/10

## Idea description

Add a geometry-guided infill operator to a population optimizer used for black-box neural-network tuning. Fit a local Jacobian from recent parameter perturbations and validation-residual vectors, generate a damped Gauss-Newton candidate for exploitation, and sample exploratory candidates in the same Jacobian-derived metric. The host optimizer retains selection, population survival, covariance adaptation, and its total evaluation budget; only a configurable fraction of new candidates is replaced by geometry-guided infill.

## Mathematical statement

The paper identifies the inverse local Gauss-Newton metric T=(J^T J+lambda I)^(-1), where J is the Jacobian of the locally modeled residual vector with respect to the search variables and lambda>0 is damping. Let z in R^p be the black-box tuning vector, r(z) in R^m a residual vector formed from validation losses on m fixed shards, and (z_i,r_i) recent evaluations. Estimate J by weighted ridge regression around the incumbent z_0: minimize the weighted squared linearization error plus eta times the squared Frobenius norm, where w_i are locality weights and eta>0 stabilizes underdetermined fits. The resulting Gauss-Newton residual-model step is delta_GN=-(J_hat^T J_hat+lambda I)^(-1)J_hat^T r_0. Exploration uses the same metric, with a random direction transformed by T^(1/2), so directions of low local residual sensitivity receive larger steps. Proposition 1 states that a covariance basis fixed independently of J cannot in general align its exploration metric with this inverse-Hessian approximation; the transferable property is therefore online, trace-based metric adaptation.

## Key formulas

- $$T=(J^{\top}J+\lambda I)^{-1}$$
- $$\hat J=\arg\min_J\sum_i w_i\left\|r_i-r_0-J(z_i-z_0)\right\|_2^2+\eta\|J\|_F^2$$
- $$\delta_{\mathrm{GN}}=-(\hat J^{\top}\hat J+\lambda I)^{-1}\hat J^{\top}r_0$$
- $$\delta_{\mathrm{explore}}=\rho\,\frac{T^{1/2}u}{\|T^{1/2}u\|_2},\qquad u\sim\mathcal N(0,I),\quad T=(\hat J^{\top}\hat J+\lambda I)^{-1}$$

## Implementation notes

Integrate this at the candidate-generation stage of CMA-ES or differential evolution, not inside backpropagation. Let the search vector z contain 10–100 black-box neural tuning variables such as LoRA adapter scales, per-layer learning-rate multipliers, clipping thresholds, quantization scales, dropout rates, or compact architecture choices. Evaluate each candidate on the same m validation shards and store the residual vector r_i, for example normalized shard losses relative to a fixed baseline. Maintain a FIFO trace of K evaluations with K at least p+m. At each generation, choose the incumbent z_0 and residual r_0, compute Delta z_i=z_i-z_0, and fit the ridge Jacobian J_hat using weights w_i=exp(-||Delta z_i||^2/h^2). Form A=J_hat^T J_hat+lambda I and solve A delta=-J_hat^T r_0 using Cholesky or conjugate gradients; do not explicitly invert A. Clip delta to trust radius rho, project it to legal parameter bounds, and insert it as one exploitation candidate. For exploration, sample u from a standard Gaussian and obtain T^(1/2)u by solving with the Cholesky factor of A, then normalize and scale by rho. Fill remaining slots with the unchanged host sampler and let the host select survivors. Increase lambda or shrink rho when predicted and observed residual decreases disagree; disable infill when the Jacobian condition estimate is poor. Estimate h, lambda, rho, and the infill fraction empirically, while computing J_hat, A, the Gauss-Newton step, and metric-shaped samples from the stated formulas. First test on CIFAR-10 using a frozen small ResNet and black-box tuning of 20 LoRA, quantization, and schedule variables, with CMA-ES as baseline. Use equal validation-evaluation budgets and measure best accuracy versus evaluations, wall-clock overhead, prediction error, and ablations removing either exploitation or exploration. Success is faster accuracy improvement at equal evaluations, at least 10% fewer evaluations to reach a fixed accuracy, and no severe failures on ill-conditioned local fits.

## Verification

- Status: beats_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: helps
- Verdict: Built Jacobian fitting, damped Gauss–Newton exploitation, and inverse-metric exploration infill in experiment.py. The exact linear check recovered the Jacobian with relative error 5.3e-11, reduced residual norm to 30.8% of its initial value, and confirmed inverse-metric eigenvalue alignment. In the paired 8-seed toy benchmark, guided infill reached mean loss 2.99e-32 versus 2.32 for the control after 500 evaluations, with much faster improvement; this is promising toy evidence, not neural-tuning validation.

### Mechanism check

- Verdict: Built Jacobian fitting, damped Gauss–Newton exploitation, and inverse-metric exploration infill in experiment.py. The exact linear check recovered the Jacobian with relative error 5.3e-11, reduced residual norm to 30.8% of its initial value, and confirmed inverse-metric eigenvalue alignment. In the paired 8-seed toy benchmark, guided infill reached mean loss 2.99e-32 versus 2.32 for the control after 500 evaluations, with much faster improvement; this is promising toy evidence, not neural-tuning validation.
- Confidence: 8/10
- Limitations: The experiment uses an 8-dimensional synthetic nonlinear residual function, not CIFAR-10, a neural network, LoRA, quantization variables, CMA-ES, or differential evolution. The control is isotropic random incumbent perturbation rather than a full CMA-ES implementation. Wall-clock overhead, prediction-error adaptation, ablations, and broad ill-conditioning stress tests were not evaluated.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 8/8
- Benchmark verdict: idea better (significant)

## Artifacts

- [bench_experiment.py](https://synthcore.org/code/1169/bench_experiment.py)
- [bench_report.json](https://synthcore.org/code/1169/bench_report.json)
- [experiment.py](https://synthcore.org/code/1169/experiment.py)
- [report.md](https://synthcore.org/code/1169/report.md)
- [report_bench_2026-09-03T010725.md](https://synthcore.org/code/1169/report_bench_2026-09-03T010725.md)
- [results.json](https://synthcore.org/code/1169/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1169)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.